Gradient Estimate on the Neumann Semigroup and Applications

TL;DR

This paper establishes a sharp gradient estimate for the Neumann semigroup on compact domains, leading to bounds on the heat kernel gradient and applications in harmonic analysis and PDE regularity.

Contribution

It provides the first sharp upper bound for the gradient of the Neumann semigroup on smooth or convex domains, with applications to heat kernel estimates and PDE analysis.

Findings

Sharp gradient bound for Neumann semigroup established

Gaussian type upper bound for Neumann heat kernel derived

Applications to Hardy spaces, Riesz transforms, and PDE regularity demonstrated

Abstract

We prove the following sharp upper bound for the gradient of the Neumann semigroup $P_t$ on a $d$-dimensional compact domain $\OO$ with boundary either $C^2$-smooth or convex: $$\|\nn P_t\|_{1\to \infty}\le \ff{c}{t^{(d+1)/2}},\ \ t>0,$$ where $c>0$ is a constant depending on the domain and $\|\cdot\|_{1\to\infty}$ is the operator norm from $L^1(\OO)$ to $L^\infty(\OO)$. This estimate implies a Gaussian type point-wise upper bound for the gradient of the Neumann heat kernel, which is applied to the study of the Hardy spaces, Riesz transforms, and regularity of solutions to the inhomogeneous Neumann problem on compact convex domains.